Peak-to-Average Power Ratio (PAPR)

Peak-to-Average Power Ratio (PAPR) Wireless Information Transmission System Lab Institute of Communications Engineering National Sun Yat-sen University 2011/07/30 王森弘

Multi-carrier systems The complex baseband representation of a multicarrier signal consisting of N subcarriers is given by where f N 1 1 j2 nft xt X n e,0 t T N n0 is the subcarrier spacing In OFDM systems, the subcarriers are chosen to be orthogonal (ie, f 1 T) 2

An example of the time-domain signals with 64 subcarriers Amplitude 035 03 025 02 015 01 005 0 0 10 20 30 40 50 60 70 Sample Index 3

The effect of high PAPR Due to the large number of sub-carriers in typical OFDM systems, the amplitude of the transmitted signal has a large dynamic range, leading to in-band distortion and out-of-band radiation when the signal is passed through the nonlinear region of power amplifier Although the above-mentioned problem can be avoided by operating the amplifier in its linear region, this inevitably results in a reduced power efficiency The PAPR of the transmit signal is defined as 0 max x t 0 t T PAPR T 2 1/ T x t dt 2 4

AM/AM distortion Soft limiter 5

Bandwidth regrowth 6

PAPR in discrete-time case If we sample x(t) by a sampling rate of 1/T s (the sampling period T s = T/N ), we may miss some signal peaks and get optimistic results for the PAPR For better approximating the true PAPR in the discrete-time case, we usually oversample x(t) by a factor of L, ie, the sampling rate is L/T s It was shown in [15] that an oversampling factor L=4 is sufficient to approximate the true PAPR 7

PAPR in discrete-time case X For an OFDM system with N sub-carriers, an oversampling rate of L can be achieved by inserting (L-1) N zeros in the middle of the modulated symbol vector to form a 1 LN data vector X, ie X 0, X 1,, X N / 2 1, 0,, 0, X N / 2,, X N 1 L1 N The PAPR computed from the L-times oversampled time-domain signal samples is given by PAPR max 0kLN 1 Ex k x 2 k 2 8

The CCDF of the PAPR The cumulative distribution function (CDF) of the PAPR is one of the most frequently used performance measures for PAPR reduction techniques In the literature, the complementary CDF (CCDF) is commonly used instead of the CDF itself The CCDF of the PAPR denotes the probability that the PAPR of a data block exceeds a given threshold From the central limit theorem, the real and imaginary parts of the time domain signal samples follow Gaussian distributions, assuming each distribution with a mean of zero and a variance of 05 for a multicarrier signal with a large number of subcarriers 9

The CCDF of the PAPR Hence, the amplitude of a multicarrier signal has a Rayleigh distribution, while the power distribution becomes a central chisquare distribution with two degrees of freedom The CDF of the instantaneous power of a signal sample is given by F z 1 exp z The CCDF of the PAPR of a data block with Nyquist rate sampling is derived as 1 P PAPR z P PAPR z This expression assumes that the N time domain signal samples are mutually independent and uncorrelated 1F z 1 1 exp z N N 10

PAPR Reduction Methods Distortion Clipping Companding Distortionless Selected Mapping (SLM) Partial Transmit Sequence (PTS) Others Active Constellation Extension (ACE) Tone Reservation (TR) 11

Clipping The simplest way to reduce the PAPR The peak amplitude becomes limited to a desired level Clipping y n xn, xn A A exp{ j arg( xn)}, xn A Clipping Ratio CR X A 20log X db : RMS value of xn 12

Clipping By distorting the OFDM signal amplitude, a kind of selfinterference introduced that degrades the BER Nonlinear distortion increases out-of-band radiation 13

Companding 14

Companding 15

Selected mapping (SLM) Serial-toparallel conversion of user bit stream Coding & Interleaving Mapping A (1) P (2) P P ( U ) (1) A (2) A ( U ) A Optionally: Differential encoding Optionally: Differential encoding Optionally: Differential encoding IDFT IDFT IDFT (1) a (2) a ( U ) a Selection of a desirable symbol a Bit source If necessary: Side information 16

Selected mapping (SLM) 1 A set of U markedly different, distinct, pseudo-random but fixed vectors P (u) ( u ) = [P (u) 0,, P (u) ( u) j n ( u) N-1 ], with Pn e, n [0,2 ), 0 n N, 1 u U must be defined 2 The subcarrier vector A is multiplied subcarrier-wise with each one of the U vectors P (u) ( u) ( u, then resulting to component A ) n An Pn, 0 n N, 1 uu 3 Then all U alternative subcarrier vectors are transformed into time ( u) ( u) domain to get a IDFT{ A } and finally that transmit sequence ( u) a a with the lowest PAPR is chosen For implementation, the SLM technique needs U IDFT operations, and the number of required side information bits is log 2 U, y denotes the smallest integer that exceed y 17

PAPR reduction performance of SLM N = 256, L = 4, 16-QAM, P { 1, 1, j, j} ( u) n 18

Partial transmit sequence (PTS) Serial-to-parallel conversion of user bit stream Coding & Interleaving Mapping (1) A (2) A IDFT IDFT (1) a (2) a (1) b (2) b + a Subblock partationing Optionally: Differential encoding ( M ) A IDFT ( M ) a ( M ) b Bit source Pack value optimization If necessary : side information 19

Partial transmit sequence (PTS) 1 In this scheme, the subcarrier vector A is partitioned into M pairwise ( m disjoint subblocks ) ( m) A, 1 mm All subcarrier positions in A which are already represented in another subblock are set to zero, so that M ( m) A A m1 ( m) 2 We introduce complex-valued rotation factors b e, ( m) [0,2 ), 1 m M, and μ is index of all phase rotation of Peak value optimization Enabling a modified subcarrier vector A m1 ( m) ( m) which represents the same information as A, if the set (as side information) is known for each μ M b A j ( m) ( m b ),1 m M 20

Partial transmit sequence (PTS) 3 To calculate a IDFT{ A}, the linearity of the IDFT is exploited Accordingly, the subblocks are transformed by M separate and parallel N-point IDFTs, yielding M M ( m) ( m) ( m) ( m) b IDFT b m1 m1 a { A } a 4 Based on them a peak value optimization is performed by suitably ( m) choosing the free parameters b such that the PAPR is minimized ( for m ) b It should be noted, that PTS can be interpreted as a structurally modified special case of SLM 21

Partial transmit sequence (PTS) In general, the selection of the phase factors is limited to a set with a finite number of elements to reduce the search complexity The set of allowed phase factors is written as j 2 n / W P e n 0,1,, W 1, where W is the number of allowed phase factors (1) In addition, we can set b 1 without any loss of performance M 1 Hence, W sets of phase factors are searched to find the optimum set of phase factors PTS needs M IDFT operations for each data block, and the number M 1 of required side information bits is log 2 W 22

PAPR reduction performance of PTS M 1 N = 256, L = 4, 16-QAM, exhausted research for W=2 means [+1,-1], W=4 means [+1, -1, +1j, -1j] W 23

Active Constellation Extension (ACE) In this technique, some of the outer signal constellation points in the data block are dynamically extended toward the outside of the original constellation such that the PAPR of the data block is reduced Im Im a 1 Re Re QPSK 24 16-QAM

Active Constellation Extension (ACE) clipping X IFFT x x FFT X x n x n, x n A j n Ae, x n A Iteration? 25 PAPR

PAPR reduction performance of ACE N = 256, L = 4, QPSK, A = 486 db 26 PAPR

Tone Reservation (TR) The transmitter reserves a small number of unused subcarriers These subcarriers are referred to as peak reduction carriers (PRCs) Since PRCs do not carry data, this increment induces a severe degradation of system s power efficiency In general, there are two approaches to reduce the PAPR in the TR technique The first is to select the PRC indices for the TR technique to be used in reducing the PAPR The second is to design the proper values on these PRCs to generate an optimal peak-canceling signal that minimizes the PAPR of a transmitted OFDM signal 27

Tone Reservation (TR) X X n, n C Cn, n n n c 28

Reference [1] S H Han and J H Lee, An overview of peak-to-average power ratio reduction techniques for multicarrier transmission, IEEE Wireless Commun, vol 12, pp 56-65, Apr 2005 [2] T Jiang and Y Wu, An overview: peak-to-average power ratio reduction techniques for OFDM signals, IEEE Trans Broadcast, vol 54, no 2, pp 257-268, Jun 2008 [3] J Armstrong, Peak-to-average reduction for OFDM by repeated clipping and frequency domain filtering, IET Electron Lett, vol 38, no 5, pp 246 247, Feb 2002 [4] X Li and L J Cimini, Effects of clipping and filtering on the performance of OFDM, IEEE Commun Lett, vol 2, no 5, pp 131 133, May 1998 [5] X Huang, J H Lu, J L Zheng, K B Letaief, and J Gu, Companding transform for reduction in Peak-to-Average power ratio of OFDM signals, IEEE Trans Wireless Commun, vol 3, no 6, pp 2030 2039, Nov 2004 [6] T Jiang and G Zhu, Nonlinear companding transform for reducing peak-to-average power ratio of OFDM signals, IEEE Trans Broadcast, vol 50, no 3, pp 342 346, Sep 2004 [7] S S Yoo, S Yoon, S Y Kim, and I Song, A novel PAPR reduction scheme for OFDM systems: selective mapping of partial tones (SMOPT), IEEE Trans Consum Electron, vol 52, no 1, pp 40 43, Feb 2006 [8] M Breiling, S H Muller, and J B Huber, SLM peak-power reduction with explicit side information, IEEE Commun Lett, vol 5, no 6, pp 239 241, June 2001 29

Reference [9] S G Kang, J G Kim, and E K Joo, A novel subblock partition scheme for partial transmit sequence OFDM, IEEE Trans Commun, vol 45, no 9, pp 333 338, Sep 1999 [10] A Ghassemi and T A Gulliver, A low-complexity PTS-based radix FFT method for PAPR reduction in OFDM system, IEEE Trans Signal Process, vol 56, no 3, pp 1161 1166, Mar 2008 [11] B S Krongold and D L Jones, PAR reduction in OFDM via active constellation extension, IEEE Trans Broadcast, vol 49, no 3, pp 258 268, Sep 2003 [12] A Saul, Generalized active constellation extension for peak reduction in OFDM systems, in Proc 2005 IEEE International Conference on Communications (IEEE ICC 2005), Seoul, Korea, Sep 2005, vol 3, pp 1974 1979 [13] J Tellado, Peak to average power reduction for multicarrier modulation, PhD dissertation, Stanford University, 2000 [14] L Wang and C Tellambura, Analysis of clipping noise and tone-reservation algorithms for peak reduction in OFDM systems, IEEE Trans Veh Technol, vol 57, no 3, pp 1675 1694, May 2008 [15] C Tellambura, Computation of the continue-time PAR of an OFDM signal with BPSK subcarriers, IEEE Commun Lett, vol 5, no 5, pp 185 187, May 2001 30